25 citations · 52 across the 10 of their papers we have counts for
15 papers · 1 filter
Refinements of Artin's primitive root conjecture
Leo Goldmakher, Greg Martin, Paul Péringuey
A famous conjecture of Artin asserts that any integer that is neither nor a square should be a primitive root (mod ) for a positive proportion of primes . Moreover,…
A converse to a theorem of Gauss on Gauss sums
Jonathan W. Bober, Leo Goldmakher
In this note we prove (under mild hypotheses) that is a nontrivial character of if and only if the Fourier transform of has magnitude 1 somewhere in $\mathbb…
Khovanskii's theorem and effective results on sumset structure
Michael J. Curran, Leo Goldmakher
A remarkable theorem due to Khovanskii asserts that for any finite subset of an abelian group, the cardinality of the -fold sumset grows like a polynomial for all suffi…
Improving the Burgess bound via Polya-Vinogradov
Elijah Fromm, Leo Goldmakher
We show that even mild improvements of the Polya-Vinogradov inequality would imply significant improvements of Burgess' bound on character sums. Our main ingredients are a lower bo…
Refinements of Lagrange's four-square theorem
Leo Goldmakher, Paul Pollack
A well-known theorem of Lagrange asserts that every nonnegative integer can be written in the form , where . We characterize the values…
Mock characters and the Kronecker symbol
Jean-Paul Allouche, Leo Goldmakher
We introduce and study a family of functions we call the "mock characters". These functions satisfy a number of interesting properties, and of all completely multiplicative arithme…
The frequency and the structure of large character sums
Jonathan Bober, Leo Goldmakher, Andrew Granville +1
Let denote the maximum of for a given non-principal Dirichlet character , and let denote a point at which the maximum is attained. In t…
Pólya-Vinogradov and the least quadratic nonresidue
Jonathan Bober, Leo Goldmakher
It is well-known that cancellation in short character sums (e.g. Burgess' estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short characte…
Large even order character sums
Leo Goldmakher, Youness Lamzouri
A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an ana…
L-functions with n-th order twists
Valentin Blomer, Leo Goldmakher, Benoit Louvel
Let K be a number field containing the n-th roots of unity for some n > 2. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central value…
A quadratic large sieve inequality over number fields
Leo Goldmakher, Benoit Louvel
We formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basi…
The distribution of the maximum of character sums
Jonathan Bober, Leo Goldmakher
We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal mag…
Lower bounds on odd order character sums
Leo Goldmakher, Youness Lamzouri
A classical result of Paley shows that there are infinitely many quadratic characters whose character sums get as large as ; this implies that a con…
Multiplicative mimicry and improvements of the Polya-Vinogradov inequality
Leo Goldmakher
We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation u…
Character sums to smooth moduli are small
Leo Goldmakher
Recently, Granville and Soundararajan have made fundamental breakthroughs in the study of character sums. Building on their work and using estimates on short character sums develop…